In [1]}, Geng and Nair resolved this converse proof of Gaussian optimality by exploiting the factorization of concave envelopes. Nevertheless, it seems indirect to extend this method to multi-receiver case. In this section, we construct the monotone path in tensorized probability space, and recover the same result as in [1]}. Thus, Our approach gives an alternative converse based on perturbation arguments such as [3]}.
